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by shovan85 » Wed Nov 03, 2010 10:39 am
nickhar130 wrote:How many integers between 250 and 300, inclusiv, can be evenly divided by NEITHER 3 NOR 5?

a) 33
b) 28'
c) 27
d) 25
e) 24

is there a trick to this, please advise?
Yes!!! Of course there is a trick. ;)

What is the first multiple of 3 between 250 and 300? Its 252.
What is the last multiple of 3 between 250 and 300? Its 300.

How many are divisible by 3 between 252 and 300?
(300-252)/3 = 48/3 = 16 but it excludes 252. Hence 17.


What is the first multiple of 5 between 250 and 300? Its 250.
What is the last multiple of 5 between 250 and 300? Its 300.

How many are divisible by 3 between 252 and 300?
(300-250)/5 = 50/5 = 10 but it excludes 250. Hence 11.

But wait there are some integers which are multiple of 3 and 5. Thus we need to discard the integers which are multiple of LCM[3,5] = 15.

What is the first multiple of 15 between 250 and 300? Its 255.
What is the last multiple of 15 between 250 and 300? Its 300.

How many are divisible by 15 between 255 and 300?
(300-255)/15 = 45/15 = 3 but it excludes 255. Hence 4.

So, Total = 11 +17 -4 = 24

Total numbers between 250 and 300 is 300-250 + 1 = 51

Not divisible by 3 and 5 = 51 - 24 = 27
Last edited by shovan85 on Wed Nov 03, 2010 10:47 am, edited 1 time in total.
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by Rahul@gurome » Wed Nov 03, 2010 10:44 am
nickhar130 wrote:How many integers between 250 and 300, inclusiv, can be evenly divided by NEITHER 3 NOR 5?

a) 33
b) 28'
c) 27
d) 25
e) 24

is there a trick to this, please advise?
Required number of integers = Total numbers of integers - Number of integers divisible by 3 - Number of integers divisible by 5 + Number of integers divisible by 15

The term "Number of integers divisible by 15" is added because numbers like 255, 270, ... 300 are divisible by both 3 and 5. So they have been discarded twice. To incorporate them we must add the number of integers that are divisible by both 3 and 5.

Now, total number of integers between 250 and 300 inclusive = 300 - 250 + 1 = 51
Now, total number of integers between 250 and 300 inclusive that are,
  • (1) Divisible by 3 = (300 - 250)/3 + 1 = 17
    (2) Divisible 5 = (300-250)/5 + 1 = 11
    (3) Divisible by 15 = (300-250)/15 + 1 = 4
Note: The integer part of the result of the division is accounted only. This is because the number of integers itself must be an integer. 1 is added to account for the inclusion of 300.

Therefore, number of integers between 250 and 300, inclusive, that can be evenly divided by NEITHER 3 NOR 5 = 51 - 17 - 11 + 4 = 27

The correct answer is C.
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by Rahul@gurome » Wed Nov 03, 2010 10:50 am
shovan85 wrote:...

IMO
E
@shovan85: The question asks for How many integers between 250 and 300, inclusive, can be evenly divided by NEITHER 3 NOR 5?

Thus, the answer is not 24. It's 51 - 24 = 27.

How was your test by the way? :)
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by nickhar130 » Wed Nov 03, 2010 10:54 am
thank you all. Explanation is very clear.
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